A second order numerical scheme for nonlinear Maxwell’s equations using conforming finite element

Abstract

In this paper, we study and analyze a second order numerical scheme for Maxwell’s equations with nonlinear conductivity, using the Nédelec Finite Element Method (FEM). A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. The curl-conforming nature of the Nédelec element assures its divergence-free property. In turn, we present the linearized stability analysis for the numerical error function to obtain an optimal L2 error estimate. In more details, an O(τ2+hs) error estimate in the L2 norm yields the maximum norm bound of the numerical solution, so that the convergence analysis could be carried out at the next time step. A few numerical examples in the transverse electric (TE) case in two dimensional spaces are also presented, which demonstrate the efficiency and accuracy of the proposed numerical scheme.